- Email: [email protected]
Fatigue crack growth in welded beams
Engineering
Frocrrrre Mrchnnics.
1973. Vol. 5. pp. 415-429.
Pergamon Press.
Printed in Great Bntain
FATIGUE CRACK GROWTH IN WELDED BEAMS MANFRED A. HlRTt Swiss Federal Institute of Technology, Institute for steel construction, Lausarme, Switzerland and JOHN W. FISHER Director Fritz Engineering Lab., Lehigh University. Bethlahem, Pa., &s&act-The fatigue behavior of welded steel beams is evaluated using the fracture mechanics concepts of stabk ctack growth. A fracture mechanics model for cracks originating from the pores in the web-to-flange t&t weld is developed. Estimates of the stress-intensity factor are made that numerically describe the initial flaw condition. With the final crack size known, a theoretical crack-growth equation was derived from the fatigue test data of the welded beams. The derived relationship compares well with actual crack-growth measurements on a welded beam and availabk data from crack growth specimens. The regime of crack growth where most of the time is spent growing a fatigue crack in a structural ekment is shown to correspond to growth rates below 109n. per cyck. Little experimental crack growth data is available at this level. It is concluded that the fracture mechanics concepts can be used to analyze fatigue behavior and to rationally evaluate the m&r variables that intluence the fatigue life of welded beams.
NOTATION a material related constant in ctack-growth equation elastic stress-intensity factor for a crack; (ksi din.) number of applied sttess cycks number ofcycles mquired for a crack to grow from size 0, to size a, nominal applied stress in the extreme fibre of the tension ftange stress nulge crack sixe; crack-radius for penny-shaped crack, half crack-width for tunnel crack or through-thethickness crack. crack-depth for surface crack, crack-radius for corner crack, minor half-axis for eBipticaicrack equivalent radius of a penny-shaped crack that provides the same K-factor estimated for the arbitrarily shapedcrack tinal crack-size initial crack-size (for integration interval) final crack-size (for integration interval) major half-axis for an elliptical crack nondimensional geometry correction factor for stress-intensity factor K exponent of crack-growth equation; slope of equation in log-log transformation (n/2) - 1 stress-intensity-factor range stress tange relevant for the determination of the stmss-intensity-factor range stress applied sufsciently away from the crack-tip
INTBODUCTION THIS PAPER evaluates the fatigue behavior of welded steel beams without attachments using the fracture mechanics concepts of stable crack growth. Earlier studies had indicated that various flaw conditions existed in welded beams. Both internal and external weld defects in the web-to-flange fillet weld were evaluated and discussed in detail in 141,[91and 1101.Internal pores caused by entrapped gas in the weldment are known to form a common defect. These gas pores and their effect on the fatigue behavior of welded beams are discussed in this paper. A fracture mechanics model for cracks originating from the pores in the web-toflange tillet weld is developed. Estimates of the stress-intensity factor are made that numerically describe the initial flaw condition. With the initial and tlnal crack size tFormerly Research Assistant. Fritz Lab.. Lehigh Univ., Bethlehem, Pa. 415
M. A. HIRT and J. W. FISHER
416
known, a theoretical crack-growth equation was derived from the fatigue test data of the welded beams. The experimental data of 56 plain-welded beams covering three grades of steel-ASTM A36, A44 1 and A5 14 were presented in [43 and [9] together with an analysis of the test variables. Fracture toughness was not a significant factor in the determination of crack propagation and the fatigue life of the beams tested. Net-section yielding generally terminated fatigue testing. Hence, no emphasis was placed on this aspect of failure as it was not relevant to this study. Fracture toughness can become a problem with low toughness steel and under adverse environmental and load conditions. These limitations were not the subject of this investigation. The main objective of this paper is to demonstrate the validity and applicability of fracture mechanics concepts of fatigue crack growth to the fatigue behavior of actual structural elements containing real flaws. It is also shown that the AK-values responsible for crack growth in these beams are well below the AK-values normally obtained during crack-growth testing. A theoretical crack-growth equation was derived from the fatigue life data of welded beams and compared with available crack-growth data. FATIGUE STBENG’I’H OF WELDED BEAMS WITH INTEBNAL WELD DISCONTINUmES It was observed that cracks in the welded beams always originated from a discontinuity (flaw) in the web-flange fillet weld unless a severe notch existed at the flame-cut flange tip. More than 180 cracks were found in the plain-welded beams reported in [4] and [9]. Seventy-five of these cracks were cut open for fractogtaphic examination. It was found that about 80 per cent of the cracks had originated from porosity caused by the entrapment of gas. In general, these cavities were completely inside the weld. The gas pores appeared on the fracture surface as rounded cavities with a smooth and shiny surface as illustrated in Fig. 1. A few pipe or blow holes extended to the surface of the weld.
/
Log N = 10.3528-
.
From
o
From Other Defects
e
At Stiffener or Attachment. no Visible Crocks in Longitudind Weld
Pomsity
e
2.9844
e
Log
e
e o-
01 No Crocks 5
IO"
5 NUMBER
I
I4
10s
5
OF CYCLES,
,,I
107
N
Fig. 2. Comparison of all welded beams failing from porosity in the fillet weld with beams failing from other defects.
Fig. 1. Exampks of porosity fbm the root ofthe kmghdhl
till&-weld. [Facing
Fig. 5. Small crack with penetration to the fillet-weld surface (-- x 11).
Fig. 6. Crack in flange-to-web junction apprtx&Gngthe extreme Abre of the tension flange (X3.6).
Fatigue crack growth in welded beams
417
All test data for beams failing from porosity in the longitudinal Glletweld are plotted by solid symbols in Fig. 2. The equation of the mean line resulting from a least squares fit to these test data is log N = IO.3528- 29844 log s,.
(1)
Previous analysis [4,9] had indicated that stress range was the dominant variable and that grade of steel did not significantly aEect the fatigue strength. The open symbols in Fig. 2 indicate plain-welded beams that failed from defects other than porosity. This included beams failing from weld repairs, stop-start positions, other weld defects, and severe notches in the flame-cut flange-tips. Generally, these beams yielded shorter lives than beams failing from porosity. when weld repairs were absent in the fillet weld of beams with stiffeners or attachments, failure occurred at these welded details since they were more critical than porosity. These beam data are also included as they revealed no visible crack growth in the longitudinal 8llet weld at lower stress ranges. An indication of a run-out level is seen at the 18 ksi stress range level. Only three data points from plain-welded beams failing from porosity were observed at that stress range. CRACK-GROWTH OBSERVATION ON A PLAIN-WRLDRD BEAM One plain-welded beam was closely observed in order to investigate crack formation and growth on the sur&es of the fillet weld, web and flange. Measurements of crack length and load were made so that crack-growth rates could be determined for the structural element and compared with the crack-growth relationship analytically derived from the S-N data of the welded beams. Beam PWB-341[4] was sekcted for this study since it had failed prematurely after 192,ooO cycles of stress from a flange-tip crack. Replicate beams tested at the same stress range of 36 ksi indicated that a crack from a gas pore could be expected to become visible within an additional 200,000 cycles. The flange-tip crack was carefully repaired and testing was resumed under the same stress conditions. The iWet welds were searched for cracks using a regular hand m&fying glass whik testing at 250 clmin. A length of more than 42 in. of weld on both sides of the web in the constant moment region was observed. Testing was interrupted periodically so that the weld could be examined under static load. Suspicious locations were further examined with a 50-power microscope. A hair& crack was detected on the weld surface shortly after an examin&ion under static load. Testing was temporarily discontinued at that time after a total of 35 1,700 applkd stress cycles. The apparent total crack-size visiWe on the weld surface was found to measure about 0.05 in. when the load was removed from the beam. This increased to O-25in. under maximum stress of 50 ksi in the extreme fibre. A reference grid was placed on the inside surfaces and the extreme fibre of the flange as well as on both faces of the web. The grids were placed close to the cross section containing the hairline crack. It was possible to estimate the crack-tip to within & of a grid line (*O*Olin.) using 50-power magni8cation. The hand m@@ing glass permitted about 4 the grid distance to be estimated (&0.025 in.). At intervals of about 18,000 cycles, testing was interrupted and the crack was measured under a statically applied maximum load. There was good agreement between the crack-length measurements under dynamic and static loading. Only the
EFM. Vol. 5 No. 2-O
418
M. A. HIRT
Fig. 3. Schematic
ofthe stages
and J. W. FISHER
of crack growth &om a pore in the met weld.
linear dimension of the crack on the fillet weld could be observed until the crack reached the extreme l&e. After about 435,000 cycles the crack was detected on the bottom surface of the flange. It yas only then observed with certainty that the crack had also penetrated through the other fiIlet weld. The increase in crack-size was very rapid after the crack had penetrated the bottom llange surface. The cross section of the flange fractured at 467,000 cycles as one cracktip reached the flange-tip. A schematic of the advancement of the crack after it had penetrated the extreme fibre is shown in Fig. 3 and indicates a transition from a pennyshaped crack with a continuous crack front in the tlange-web junction to a three-ended crack with two crack tips in the IIange and one in the web. Hence, two basic stages of growth were observed for cracks origiuating from pores in the longitudinal Pelletweld of welded beams. The first stage of growth in the flangeto-web junction from the initial crack-size up to the point where the crack reached the extreme fibre of the tlange. After penetration of the extreme &nge-fibre, the crack changed its shape rapidly to become a three-ended crack. A PKNNY4EAPED
CRACK MODEL Fractographic examination of small cracks that had originated from pores in the ftllet weld showed that many were almost perfectly circular in shape. This circular shape was found at various stages of growth up to the point where the crack had reached the extreme fibre of the flange, as illustrated in Figs. 4-6. One of the smallest cracks was discovered when examming a crack in the iillet weld. A tiny crack about 0.07~in. in diameter had originated from a very small pore in the left weld as shown in Fig. 4. The extent of the circular crack is seen from the smooth crack surface surrounding the pore, as compared to the rough appearance caused by static tearing when the cross section was opened for inspection. The small crack on the left was completely inside the Lange-to-web junction and could not be detected by inspection of the weld surface. A small crack discovered by the magnetic particle inspection method is shown in Fig. 5. The crack had initiated from the elongated pore and grown to the surface of the
Fatigue crack growth in welded beams
419
fillet weld. Almost no deviation from a circular shape is visible. This crack is about O-26in. diameter and could not be detected on the weld surface with the aid of a magnifying glass even under favorable circumstances and under sustained loading. A crack which has nearly penetrated the extreme fibre of the tension flange is shown in Fig. 6. Again, no significant disturbance of the circular shape is apparent at either side of the web or at the front approaching the surface of the flange. This crack, although quite sizeable, is not easily detected on a structural element under applied loading. The linear dimension of the crack visible on the surface of the flange-to-web junction is somewhat less than an inch. This phenomenon of a circular crack-shape,. seemingly not influenced by the free surfaces, is believed to be a consequence of the compatibility condition of the basically elastic cross section. Based on this photographic evidence, a circular disc-like crack was assumed to model the actual crack duriug growth inside the flange-web connection as was illustrated in Fig. 3. EVALUATION OF FATIGUR BRHAVIOR USING FiMTURR MECHANKS CONCEPTS The fatigue test data and fractographic observations of small fatigue cracks suggested that fracture mechanics of stable crack-growth might be useful in evaluating the observed behavior of the test beams. The stress intensity factor K introduced by Invin [ 111 describes in convenient form the influence of the stress, applied sufficiently away from the crack tip, and the crack size a. The relative size is usually expressed in terms of a correction function, f(u), where the linear dimensions of the plate, or the distance to a freeedge or surface, are introduced.
Paris [ 151 has suggested the following relationship between the rate of crack propagation and the change in the stress-intensity factor:
& ==CA&?.
(3)
This model expresses crack growth per cycle, d&N, in terms of the variation of the stress intensity factor, AK, and two material constants, C and n. For the case of a crack with constant correction factor, f(a), and subjected to constant amplitude stress, integration of equation (3) yields
where a! = (r&2)- 1. For specimens with equal initial crack-size Q, final crack-size ufi and identical boundary conditions, a theoretical value for the prediction of the life interval Nu can be expressed in terms of a new constant C’ times the applied stress EillgeAUas Nu =
C’Au-”
(5)
420
M. A. HIRT and J. W. FISHER
where
The log-log transformation of equation (5) yields a straight line of the form log N, = log C’ - n log (Au). AI’PLKATiOl’J
TO TEE PLAINWBLDED
(7)
BEAM
In order to apply the outlined concepts of fracture mechanics a reasonable characterization of the flaws was needed. A large number of cracks that origmated from porosity in the flllet weld were examined closely to establish the initial flaw condition. Flaws were photographed and enlarged to at least four times, as illustrated in Fig. 1.The dimensions of the defects were measured on the photos under lo-power magnification. Cracks were examined in beams fabricated from three diRerent grades of steel. The estimated K-values corresponding to the measured flaw dimensions are summarized in Fig. 7 in a log-log presentation of AK/Au vs the controlling crack-size a. The estimate for the defect that corresponds to the largest observed crack from each beam is indicated by a solid symbol. The estimated K-values were obtained using a circumscrii ellipse for flaws similar to the shapes illustrated in Fig. l(a) and Fig. 6. For flaws comparable to those shown in Fig, l(b) and Fig. -5, the K+stimate at the narrow transition from the circular void to the elongated pore was generally used. It was previously concluded from fractographic inspection of very small fatigue cracks (Figs. 4 and 5) that a penny-shaped crack-model described the shape of these cracks during crack growth. However, the estimated K-values for the initial flaws shown in Fig. 7 represent a collection of elliptical shapes with various &ratios. Hence, each individual defect was transformed into a penny-shaped crack with an
0.4-
0. I
0.01
A
A36
o
A441
I
I
1
I
0.02
0.03
0.04
0.05
CRACK
SIZE.
1
1
o. I”
Fig. 7. Distribution of estimated k-values and equivalent crack-radii.
I
1
0. IO
421
Fatigue crack growth in welded beams
equivalent crack-radius a, corresponding to the estimated AK/Ao-value. The sample of the resulting equivalent crack-radii is indicated at the bottom of Fig. 7. An average equivalent crack-radius a, = 0.04 in. was selected to represent the sample of measured pores. The initial crack-radius af was assumed equal to this average value. Derivation ofcrack-growth constants. The coefficients of the crack-growth equation can now be established from the equivalence of the coefficients in equations (1) and (7). Equation 7 is the theoretical prediction of the fatigue life using fracture mechanics concepts, and equation 1 expresses the statistical mean line of the relevant fatigue test data shown by solid dots in Fig. 2. The following assumptions were made to assist with the evaluation of the crack-growth constants, C and n: (1) The crack was assumed to be described by a disc-like penny-shaped crack with a constant correction factorf(u) over the interval of integration. The correction . factorsf(u) for a penny-shaped crack is 2/n= (2) The estimates of the initial and final crack-radii uj and uj were available from visual observations. The average initial crack-radius a, was assumed to equal the estimate of the equivalent crack-radius II, = O-04in. from Fig. 7 representing the sample of measured pores. The final crack-radius a, was assumed to be reached when the crack had penetrated the extreme fibre of the flange. The life remaining after this occurred was at most 10 per cent of the total fatigue life of the beam as was illustrated by the measurements given in Fig. 3. The final crack-radius was assumed to be equal to the nominal flange thickness. Experimental observations indicated this to be reasonable. (3) It was assumed that all three grades of steel could be represented by the same crack-growth equation. This assumption was based on results of tests on the welded beams. It was shown in Ref. 4 that grade of steel did not significantly influence the fatigue life of the beams. Equating the values of the coefficients for the mean regression curve [equation (111 n and C’ [equation (7)1, and substituting the crack-radii a, and a,, and the correction factorf(u) into equation (6) yields growth constants n a 3 and C = 2.05 X IO-*9 Substitution of these constants into equation 3 numerically describes the crackgrowth rate in terms of AK: to the coe&ients
$$ = 2.05 x lo-l”AK3.
W-t
COMPARISON OF DKRIVKD CRACK-GROWITI KQUATION WITR
MWSURRD CRACK-GROWTH RATES Equation 8 as derived from the mean regression curve fatigue test data is plotted in Fig. 8 and compared with the various stages of crack growth in the test beam illustrated in Fig. 3. The straight line estimate for crack growth as a penny-shaped crack from crack initiation until penetration of the extreme fibre is indicated. Also shown are the data points for the measured crack-growth rates on the inside and outside surfaces of the flange for growth as a three-ended crack.
of
IThe dimensions used for crack-growth rates, da/dN, were h/cycle, AK, was ksi 6.
and the stress-intensity factor range,
422
M. A. HIRT and J. W. FISHER
It is apparent from Fig. 8 and the schematic shown in Fig. 3 that most of the life is spent growing the crack from its initial equivalent flaw radius, a, = 0.04 in, to its penetration of the extreme fibre of the flange, a, = O-375in. The corresponding range of AK for the test beam was between 8 ksi 6 and 25 ksi V’& under a constant amplitude stress range of 36 ksi while 435,000 cycles elapsed. The transition from a penny-shaped crack to a three-ended crack only required about 16,000 cycles. An additional 10,000 cycles were required to grow the crack large enough to cause netsection yielding. Final failure terminated testing at 467,000 cycles. It is also apparent from Fig. 8 that most of the life was spent while growth occurred in a region of small AK. This is particularly true for lower applied stress ranges as illustrated in Fig. 9. The AK-regions applicable to growth as a penny-shaped crack are indicated for the test beams subjected to the stress ranges used in this study. Crack initiation took place at AK-values below 10 ksi 6. in all test beams. Most of the life was spent at growth rates smaller than lO+ in/cycle. The derived crack-growth relationship given by equation 8 is also compared with data from crack-growth specimens in Fig. 9. It was necessary to extrapolate the curve into the higher growth rate region to compare with available growth-rate data. The extrapolated curve falls within the scatterband reported by Crooker and Langel31. This scatterband contains data from tests on carbon and low-alloy steels with yield strengths between 34 ksi and 127 ksi. This is comparable to the yield strengths of the steel beams. A conservative upper bound for growth rates on ferrite-pearlite steels was proposed by Barsom [ 1I as
$$ = 3.6 x 10-‘“pK3.
(9)
This relationship is parallel to equation (8).
k
l
Crack on
Inside
A
Crackon
Extreme
Surface Fibfe
10-S
I
10-7
I
10-s CRACK-GROWTH
I
10-s RATE,
da/dN,
L
IO-' in
/cycle
Fig. 8. Various stages of crack growth and corresponding growth rates for a crack in a plainwelded beam starting from a pore in the flllet weld.
Fatigue crack growth in welded beams
423
Maddox (hf. 13)
E lJY
I
I 10-a
I IO-’
10-a
CRACK-GROW’TII
RATE.
I 10-s
I UP
doldN , in./CyEyclr
Fig. 9. Ranges for crack growth as a penny-shaped crack. Scatterbands and an upper bound estimate from other crack growth studies.
Since much of the growth in the plain-welded beams took place in the weld metal and heat affected zone, a comparison of the theo&& curve with Maddox’s data [ 133 is relevant. An approximate envelope for Maddox’s data on four diRerent weld metals is shown. Three had about equal yield strength (67 ksi), the fourth had a higher yield point equal to 90 ksi. Also included in the scatterband are test data for a simulated heat affected zone in mild steel material with the same yield strength. The correlation with these crack-growth data is good. It is apparent from Fig. 9 that the Croaker-Lange scatterband does not cover the critical region of interest for plainwelded beams. Most studies of crack-growth rates have been limited to larger AK-values and higher crack-growth rates because of the difficulties encountered in the slow growth region. However, the Woretical curve [Eq. @)I extrapolatedinto the higher AK-regions shows the same general trend reported by others on basic crack-growth specimens (Fig. 9). The &or&al curve is just above the crack-growth data and underestimates their growth rate. This is surprising since the penny-shaped crack assumes the best condition for the crack in the welded beams and neglects the influence of free surfaces. This underestimate in gt~~wthrate may be due to crack initiation or an overestimate of the stress intensity. The exponent, n, of the predicted crack-growth equation was about equal to 3-00 [Eq. @)I. It represents the slope of the fatigue test data on plain-weldedbeams fabricated from three grades of steel. Crooker and Lange [3] observed from a review of the literature that the value of the slope n fell between 2 and 4 for a large range of steels. Gurney [6] reported the slope of the curve to be a linear function of yield stress of the material. A general trend observed in crack-growth studies is for the value of the exponent, n, to decrease with increasing yield strength of the material. This trend was not as pronounced in the welded-beam study. The exponent n did not appear to vary greatly from 3. Careful evaluation of the coefficients n and C is needed for a wider range of rates of growth. Most of the data generally used to fit the straight line approximation only
424
M. A. HIRT and J. W. FISHER
extend over a small range of AK. In other cases, data points at the extremes cause rotation of the line. Substantial over- or under-estimates of the fatigue life of a structural component might result if crack-growth relationships are used to extrapolate beyond the range of the test data. Very slow growth. Johnson and Paris [ 121have suggested that a threshold exists for crack growth. A recent investigation by Paris [ 161on ASTM 93 10 steel has provided more information on this phenomenon of very slow growth. The test specimens were precracked and growth data for small AK-values obtained. The value of AK was then reduced stepwise and observations made on the relative crack-growth rates until a “threshold value” was reached. When AK was increased in steps, the rate of growth was comparable to the rates obtained by previous overloading. An approximate mean line fit to Paris’ data is compared in Fig. 10 with equation (8). A drastically reduced growth rate for AK = 5.2 ksi 6 is observed. Similar deviations from the straight line approximation are found for large AK-values indicating an increased growth-rate and apply mainly to low cycle fatigue problems. Harrison[7] has reviewed the literature for run-out data on a variety of specimens. He concluded that “for all materials with the exception of pure aluminum, cracks will not propagate if AK/E < 1 X 10-4&” He also found for a number of materials that the limit for non-propagating cracks fell between AK/E = 1.5 X lo44 and 1.8 X 10-l& Harrison’s levels for four types of steel are also shown in Fig. 10, and are , between 3.3 ksi & for mild steel and 5.3 ksi & for austenitic steel. Iqjlkence of defect-size on fatigue [email protected] life interval between two crack-sixes a( and a, can be determined from equation (4). For a given Iinal crack-size, the computed number of cycles can be expressed as a function of the initial crack-size for a given stress range, as shown by the curved lines in Fig. 11. The final crack-radius for the penny-shaped crack-model was assumed equal to the I&urge-thickness of the welded test beams. This becomes less important for larger differences between initial crackradius ai and final radius a, Alternately, the observed fatigue life of a tested beam can be used with the approp
“I
'10-s
10-e CRACK-GROWTH
10-p
lo-7
RATE,
do/dN,
10-5
in./cyti
Fig. 10. Approximate mean line of Paris’ data for very slow growth, and Harrison’s limiting AK-values for non-propagating cracks.
Fatigue crack growth in welded beams Stress
425
Ronpe, ksi
Mwaund , Qukoknt Flow - Radius %
1 2
NUMBER Fig.
I 1.
OF CYCLES
Cosnparism of the scatter of measured equivalent Aaw-sizes with t&e scatter of flawsizes derived ftom fktigue data.
riate stress-range line to estimate the equivalent initial pore-radius that caused faih,ue. The fatigue data of beams failing from porosity from Fig. 2 are replotted in this manner in Fig. 11. The resulting scatter of the equivalent pore-radii derived from the fatigue test data is shown on the right ordinate. The equivalent pore-radii determined independently from measured f&s are shown on~k~o~~eofF~ ll.Thtscrttteroftfiepwosi.tyasdtrivedfromthe~ test data (right ordinate) is seen to be similar to the scatter from the measured equivalent pore-sixes (left ordinate). Hence, if no variation is assumed in the crack-growth characteristics, this comparison suggests that the scatter in the fatigue test data is caused by the variation in the initial crack&e. This is essentially the same conclusion reached by Harrison [81for the analysis of test results from butt welds with lack of penetration defects. He concluded that the width of the scatterband probably resuhed from variations in initial radii at the defect tips. ~e~c~~ of furigue l&c using dwerertt cr~~-~e~. Additional crack-models were used to assess their influence on the prediction of the fatigue life. Figure 12 summa&es the results for the various models used. Six A441 steel beams were sekcted for the comparison. The actually observed fatigue lives of these six beams are shown by solid dots together with the mean and the 95% confidence interval for all test beams failing from porosity. The defects that caused crack growth and f’ailure of the six beams were examined. They were assumed to be characterixed by a circumscrii eilipse with half-axes a and b. The following models were used to estimate the fatigue life of each individual beam using the known dimensions of the circumscribed ellipse, and empioying crack-growth equation (8): Model (a) me open circles in Fig. 12 correspond to the life estimates for the penny-shaped crack with equivalent crack-radius. The equivalent crack-radius was computed from the equivalence of the K-value with the circumscribed elliptical crack.
M. A. HIRT and J. W. FISHER
426
/ /
Mean Life and ConfidenceInterval from 011 Welded Beams Failinq from Porosity (Fig. 2 1
(a)
o Equivalent Cmch - Size
(b)
I- Circumscribed and -( Inscribed Circle
Made1 (c) A Elliptical Crock-Model ‘L,-’
51 IO’
l I1
..,I
IO‘ . 5 NUMBER OF CYCLES
Observed Fotiaue Life 1
5
I
.
..I
IO’
Fig. 12. Prediction of fatigue life using various crack-models.
Model (b) Penny-shaped cracks were assumed to inscribe and circumscribe the ellipse. The shortest lie estimates resulted from the circumscribed crack and the longest from the inscribed crack. These two estimates constitute “upper and lower bounds” and are indicated by the horizontal T-symbols in Fig. 12. Model (c) Additional life estimates were made based on the elliptical crack-model. Growth was assumed in the direction of the minor axis 2(a) with constant major axis 2(b) until the size of the circumscribed crack was reached. The computation of the increment of life from an elliptical to a circular crack was done using an available computer program[5]. Numerical integration was employed because of the variable correction factor f(a). This increment of life ivas added to the estimate for the circumscribed penny-shaped crack. The total life estimate is indicated by triangles in Fig. 12. The comparison between the estimates for each individual beam provided by the different models and the observed fatigue life permitted the following observations to be made: (1) The life estimates employing the equivalent penny shaped crack-radius [model (a)] resulted in slightly shorter lives than the estimates that assumed the elliptical crack first to grow to a circular shape [model (c)l. (2) Both of these estimates were bounded by the estimates provided by the inscribed and circumscribed initial cracks [model (b)]. (3) The observed fatigue data were also contained within the bounds from the estimates of the circumscribed and inscribed circles in all but one case. (4) Long flaws with small a/b-ratios [as illustrated in Fig. l(b)] provided the greatest deviation between the computed “lower and upper bounds.” (5) All three defects of the type shown in Fig. l(a) had their life underestimated when the equivalent initial crack-radius was used [model (a)]. The estimate improved for the elliptical model (c). (6) Most of the estimates from models (a) and (c) fell within the two limits of disper-
427
Fatigue crack growth in welded beams
sion representing the 95 per cent confidence interval of all the beam test data. The selection of a penny-shaped crack model did not introduce more scatter than was experimentally observed from fatigue test data Prediction of fatigue life using measured crack-growth rates. Maddox[l4] and Harrison [8] have demonstrated that good predictions ofthe fatigue life can be obtained from crack-growth data. Barsom’s[l] conservative estimate of crack-growth rate, equation (91, for ferrite-peariite steels was used to predict the fatigue life of the beams failing from porosity. The result is shown in Fig 13 together with the applicable welded beam test data and the respective mean line and the 95 per cent confidence interval. Also shown in Fig. 13 is the prediction based on Paris’ data [ 161,indicating a threshold level for run-out at about 23 ksi stress range. This prediction was obtained using the straight line segments shown in Fig. 10 which approximate the data from the study on very slow growth. The threshold values by Harrison[7] shown in Fig. 10 were also used to estimate run-out. Run-out tests are predicted to occur at levels as high as 20 ksi stress range for low-alloy steel, as shown by the horizontal lines in Fig. 13. The predictions based on the crack-growth data underestimate the mean fatigue life of the test beams at the higher stress range levels. This was expected from the comparison of the crack-growth data Since any assumption other than a penny-shaped crack-model for the welded beams would further reduce the life prediction based on crack-growth data, a crack-initiation period may be responsible for the slight underestimate. Other factors that could be responsible for the underestimate in life are an overestimate of the initial crack-size, an overestimate of the stress intensity, or a slower growth rate under plane-strain conditions [2]. ThescatterinthetestdatashowninFig. 13waspr&ouslyrelatedtothevariation inshape,sizeandseverityoftheporosity.othtrfactorscontributetothevariationin test data. Beams with seemingly equal defects still experience variation in the fatigue strength because of variation in the crack tip radius, crack-growth rates, and other uncontrolled variables. 100
r 95 K
Estimota
Confidence lntrrvol
from Growth Dota
From “Threshold” (Fig. IO) Low-Alloy
StnlJ
Mild Stool 5105
NUMBER Fig. 13. Comparison
I ~.~~I
I ,..,I 5
IO’
5
IO’
OF CYCLES
of a pndictcd lower bound and a mean line with mean life and scatterband from test data.
428
M. A. HIRT and J. W. FISHER
SUMMARY AND CONCLUSIONS The main findings of this study are summarized hereafter. They are based on a detailed examination of the test data, fracture surfaces, initial flaw conditions provided by the experimental work, and on the theoretical studies of stable crack growth. (1) Fractographic e~on of the initial flaw conditions revealed porosity to be the most common defect in plain-welded beams. The distribution of the size and location of the pores in the longltudlnal fillet weld was random. The welded beam fatigue data fell within a narrow scatterband when plotted as the logarithmic transformation of stress range and cycles to failure. (2) Fracture mechanics concepts provided a rational way to analyze and characterize the behavior of welded (and rolled) beams. These concepts were applied to describe numerically the “efktive” initial Saw conditions in welded beams, and to derive a crack-growth rate vs range of stress-intensity relationship from welded-beam fatigue data. (3) A penny-shaped crack was found to model crack growth from porosity in welded beams. An equivalent crack with a GO4in radius described the average pores observed in the welded beams. (4) The derived crack-growth equation exhibited the same trend as measured data from crack-growth specimens. It provided a lower estimate of the growth rate. Among other factors, this dl@erence was attributed to crack axon, a possible overestimate of the stress intensity, and slower growth in the piane-strain condition for the welded beams. (5) Available crack-growth data were shown to only cover a small region of growth rates. Extrapolation into regions outside the data could be misleading, particularly when used for fatigue life estimates. (6) Very little information is available for growth rates below 1W8in./cycle. This was found to be the region most critical for the fatigue behavior of welded and rolled beams. More than 75 per cent of the life was spent in this region growing a crack from its initial size to a visible crack. (7) It was shown that the initial flaw-size was the controlling factor for the fatigue life of welded beams. An increase in flange-thickness and larger weld-sizes should hence not permit an increase in allowable defect-size. (8) The variation of the fatigue data within the scatterband was found to be related in part to the variation in fatigue crack-growth rates or other uncontrolled variables. ~c&~~fe~ge~~f~-~ paper is based on the cxperimentai and theoretical ~v~~ns made during the course of a research progmm on the &cct of weidmcnts on the fatigue strength of steel beams. Partial support was also provided by the tXIc.c of Naval Research, Dcpartmmn of Defense, under Contract NO001468-A-514; NRO64-509. The expmimental research was performed under National Cooperative Highway Research Prngram Project 12-i’. The opinions and 6ndings expnssed or implied in this paper are those of the authors. They arc not necessarily those of the ffighway Reset&r Board, the National Academy of Sciences, the Federal Highway Administration, the Amatican Association of State Highway Gfiiciais, nor of the individual states participating in the NCHRP Program. The authors wish to express their appreciation for the helpful suggestions made by Profs. G. R. Irwin, A. W. Pcnsc. and R. W. Hertzberg on the fracture mechanics and metaUurgicat aspects of this work. Special thanks are due Karl H. Frank and Ben T. Yen for the cooperation and chaBenging discussion during the progmss of this study,
REFJXRENCES [ll J. M. Barsom, Fatigue-crack propagation in steels of various yield strengths, U.S. Steel Corp., Applied Research Laboratory, Monrocvihe, Pa.. ( 197 I ). 121 W. G. Jr. Clark, and H. E. Jr. Trout, InfIuence of temperature and section size on fatigue crack growth behavior in Ni-Mo-V Alloy Steel. Engng Frucrure Me&. 2. 107-123 (1970).
Fatigue crack growth in welded beams
429
[3] T. W. Crookcr and E. A. Lange, How yield strength and fracture toughness considerations can influence fatigue design procedures for structural steels, Welding Research Supplement, 49, (10) 488-496 October, (1970). [41 J. W. Fisher, K. H. Frank, hf. A. Hirt and B. M. McNamee, Effect of weldments on the fatigue strength of steel beams, NCHRP Report No. 102, Highway Research Board, National Academy of SciencesNational Research Council, Washington, D.C., (1970). [51 K. H. Frank and J. W. Fisher, Analysis of error in determhnng fatigue crack growth rates, Report No. 358-10, Lehigh University, Fritz Engineering Laboratory, March, (1971). 161 T. R. Gurney, An investigation of the rate of propagation of fatigue cracks in a range of steels, Members’ Report No. E18/12/68, The Welding Institute, December, (1968). PI J. D. Harrison, An analysis of data on non-propagating fatigue cracks on a fracture mechanics basis, Metal Construction and Br. Weld. .I. 2.93-98 (1970). 181 J. D. Harrison, The analysis of fatigue test results for butt welds with lack of penetration defects using a fracture mechanics approach, Weld. World, 8,168- 18 1 ( 1970). 191 M. A. Hirt, B. T. Yen, and J. W. Fisher, Fatigue strength of rolled and welded steel beams, J. Srruc. Diu. ASCE,97,1897-1911(1971). [lo] M. A. Hit?, Fatigue behavior of roiled and welded beams, Ph.D. Dissertation, Lehigh University, Department of Civil Engineering, October, ( 197 1). [ 11) G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Transactions, ASME, Series E, Vol. 24, (3) 361-364 (1957). [12] H. H. Johnson and P. C. Paris, Sub-critical flaw growth, Engng Fracture Mech. 1, (1) 3-45 (1968). 113) S. J. Maddox, Fatigue crack propagation in weld metal and heat affected zone material, Members’ Report No. E/29/69, The Welding Institute, December, (1969). I.141 S. J. Maddox, The propagation of part-through-thickness fatigue cracks analysed by means of fracture mechanics, Members’ Report No. E/30/69, The Welding Institute, December, (1969). [15] P. C. Paris, The fracture mechanics approach to fatigue, Fatigue-An interdisciplinary Approach, Proc. Tenfh Sagamore Army Materials Research Cot&, Syracuse University Press, Syracuse, N.Y., p. 107 (1964). 1161 P. C. Paris, Testing for very slow growth of fatigue cracks, Closed Loop, MTS Systems Corp., 2, 5, I l-14 (1970). (Received 4 April 1972)
Frocrrrre Mrchnnics.
1973. Vol. 5. pp. 415-429.
Pergamon Press.
Printed in Great Bntain
FATIGUE CRACK GROWTH IN WELDED BEAMS MANFRED A. HlRTt Swiss Federal Institute of Technology, Institute for steel construction, Lausarme, Switzerland and JOHN W. FISHER Director Fritz Engineering Lab., Lehigh University. Bethlahem, Pa., &s&act-The fatigue behavior of welded steel beams is evaluated using the fracture mechanics concepts of stabk ctack growth. A fracture mechanics model for cracks originating from the pores in the web-to-flange t&t weld is developed. Estimates of the stress-intensity factor are made that numerically describe the initial flaw condition. With the final crack size known, a theoretical crack-growth equation was derived from the fatigue test data of the welded beams. The derived relationship compares well with actual crack-growth measurements on a welded beam and availabk data from crack growth specimens. The regime of crack growth where most of the time is spent growing a fatigue crack in a structural ekment is shown to correspond to growth rates below 109n. per cyck. Little experimental crack growth data is available at this level. It is concluded that the fracture mechanics concepts can be used to analyze fatigue behavior and to rationally evaluate the m&r variables that intluence the fatigue life of welded beams.
NOTATION a material related constant in ctack-growth equation elastic stress-intensity factor for a crack; (ksi din.) number of applied sttess cycks number ofcycles mquired for a crack to grow from size 0, to size a, nominal applied stress in the extreme fibre of the tension ftange stress nulge crack sixe; crack-radius for penny-shaped crack, half crack-width for tunnel crack or through-thethickness crack. crack-depth for surface crack, crack-radius for corner crack, minor half-axis for eBipticaicrack equivalent radius of a penny-shaped crack that provides the same K-factor estimated for the arbitrarily shapedcrack tinal crack-size initial crack-size (for integration interval) final crack-size (for integration interval) major half-axis for an elliptical crack nondimensional geometry correction factor for stress-intensity factor K exponent of crack-growth equation; slope of equation in log-log transformation (n/2) - 1 stress-intensity-factor range stress tange relevant for the determination of the stmss-intensity-factor range stress applied sufsciently away from the crack-tip
INTBODUCTION THIS PAPER evaluates the fatigue behavior of welded steel beams without attachments using the fracture mechanics concepts of stable crack growth. Earlier studies had indicated that various flaw conditions existed in welded beams. Both internal and external weld defects in the web-to-flange fillet weld were evaluated and discussed in detail in 141,[91and 1101.Internal pores caused by entrapped gas in the weldment are known to form a common defect. These gas pores and their effect on the fatigue behavior of welded beams are discussed in this paper. A fracture mechanics model for cracks originating from the pores in the web-toflange tillet weld is developed. Estimates of the stress-intensity factor are made that numerically describe the initial flaw condition. With the initial and tlnal crack size tFormerly Research Assistant. Fritz Lab.. Lehigh Univ., Bethlehem, Pa. 415
M. A. HIRT and J. W. FISHER
416
known, a theoretical crack-growth equation was derived from the fatigue test data of the welded beams. The experimental data of 56 plain-welded beams covering three grades of steel-ASTM A36, A44 1 and A5 14 were presented in [43 and [9] together with an analysis of the test variables. Fracture toughness was not a significant factor in the determination of crack propagation and the fatigue life of the beams tested. Net-section yielding generally terminated fatigue testing. Hence, no emphasis was placed on this aspect of failure as it was not relevant to this study. Fracture toughness can become a problem with low toughness steel and under adverse environmental and load conditions. These limitations were not the subject of this investigation. The main objective of this paper is to demonstrate the validity and applicability of fracture mechanics concepts of fatigue crack growth to the fatigue behavior of actual structural elements containing real flaws. It is also shown that the AK-values responsible for crack growth in these beams are well below the AK-values normally obtained during crack-growth testing. A theoretical crack-growth equation was derived from the fatigue life data of welded beams and compared with available crack-growth data. FATIGUE STBENG’I’H OF WELDED BEAMS WITH INTEBNAL WELD DISCONTINUmES It was observed that cracks in the welded beams always originated from a discontinuity (flaw) in the web-flange fillet weld unless a severe notch existed at the flame-cut flange tip. More than 180 cracks were found in the plain-welded beams reported in [4] and [9]. Seventy-five of these cracks were cut open for fractogtaphic examination. It was found that about 80 per cent of the cracks had originated from porosity caused by the entrapment of gas. In general, these cavities were completely inside the weld. The gas pores appeared on the fracture surface as rounded cavities with a smooth and shiny surface as illustrated in Fig. 1. A few pipe or blow holes extended to the surface of the weld.
/
Log N = 10.3528-
.
From
o
From Other Defects
e
At Stiffener or Attachment. no Visible Crocks in Longitudind Weld
Pomsity
e
2.9844
e
Log
e
e o-
01 No Crocks 5
IO"
5 NUMBER
I
I4
10s
5
OF CYCLES,
,,I
107
N
Fig. 2. Comparison of all welded beams failing from porosity in the fillet weld with beams failing from other defects.
Fig. 1. Exampks of porosity fbm the root ofthe kmghdhl
till&-weld. [Facing
Fig. 5. Small crack with penetration to the fillet-weld surface (-- x 11).
Fig. 6. Crack in flange-to-web junction apprtx&Gngthe extreme Abre of the tension flange (X3.6).
Fatigue crack growth in welded beams
417
All test data for beams failing from porosity in the longitudinal Glletweld are plotted by solid symbols in Fig. 2. The equation of the mean line resulting from a least squares fit to these test data is log N = IO.3528- 29844 log s,.
(1)
Previous analysis [4,9] had indicated that stress range was the dominant variable and that grade of steel did not significantly aEect the fatigue strength. The open symbols in Fig. 2 indicate plain-welded beams that failed from defects other than porosity. This included beams failing from weld repairs, stop-start positions, other weld defects, and severe notches in the flame-cut flange-tips. Generally, these beams yielded shorter lives than beams failing from porosity. when weld repairs were absent in the fillet weld of beams with stiffeners or attachments, failure occurred at these welded details since they were more critical than porosity. These beam data are also included as they revealed no visible crack growth in the longitudinal 8llet weld at lower stress ranges. An indication of a run-out level is seen at the 18 ksi stress range level. Only three data points from plain-welded beams failing from porosity were observed at that stress range. CRACK-GROWTH OBSERVATION ON A PLAIN-WRLDRD BEAM One plain-welded beam was closely observed in order to investigate crack formation and growth on the sur&es of the fillet weld, web and flange. Measurements of crack length and load were made so that crack-growth rates could be determined for the structural element and compared with the crack-growth relationship analytically derived from the S-N data of the welded beams. Beam PWB-341[4] was sekcted for this study since it had failed prematurely after 192,ooO cycles of stress from a flange-tip crack. Replicate beams tested at the same stress range of 36 ksi indicated that a crack from a gas pore could be expected to become visible within an additional 200,000 cycles. The flange-tip crack was carefully repaired and testing was resumed under the same stress conditions. The iWet welds were searched for cracks using a regular hand m&fying glass whik testing at 250 clmin. A length of more than 42 in. of weld on both sides of the web in the constant moment region was observed. Testing was interrupted periodically so that the weld could be examined under static load. Suspicious locations were further examined with a 50-power microscope. A hair& crack was detected on the weld surface shortly after an examin&ion under static load. Testing was temporarily discontinued at that time after a total of 35 1,700 applkd stress cycles. The apparent total crack-size visiWe on the weld surface was found to measure about 0.05 in. when the load was removed from the beam. This increased to O-25in. under maximum stress of 50 ksi in the extreme fibre. A reference grid was placed on the inside surfaces and the extreme fibre of the flange as well as on both faces of the web. The grids were placed close to the cross section containing the hairline crack. It was possible to estimate the crack-tip to within & of a grid line (*O*Olin.) using 50-power magni8cation. The hand m@@ing glass permitted about 4 the grid distance to be estimated (&0.025 in.). At intervals of about 18,000 cycles, testing was interrupted and the crack was measured under a statically applied maximum load. There was good agreement between the crack-length measurements under dynamic and static loading. Only the
EFM. Vol. 5 No. 2-O
418
M. A. HIRT
Fig. 3. Schematic
ofthe stages
and J. W. FISHER
of crack growth &om a pore in the met weld.
linear dimension of the crack on the fillet weld could be observed until the crack reached the extreme l&e. After about 435,000 cycles the crack was detected on the bottom surface of the flange. It yas only then observed with certainty that the crack had also penetrated through the other fiIlet weld. The increase in crack-size was very rapid after the crack had penetrated the bottom llange surface. The cross section of the flange fractured at 467,000 cycles as one cracktip reached the flange-tip. A schematic of the advancement of the crack after it had penetrated the extreme fibre is shown in Fig. 3 and indicates a transition from a pennyshaped crack with a continuous crack front in the tlange-web junction to a three-ended crack with two crack tips in the IIange and one in the web. Hence, two basic stages of growth were observed for cracks origiuating from pores in the longitudinal Pelletweld of welded beams. The first stage of growth in the flangeto-web junction from the initial crack-size up to the point where the crack reached the extreme fibre of the tlange. After penetration of the extreme &nge-fibre, the crack changed its shape rapidly to become a three-ended crack. A PKNNY4EAPED
CRACK MODEL Fractographic examination of small cracks that had originated from pores in the ftllet weld showed that many were almost perfectly circular in shape. This circular shape was found at various stages of growth up to the point where the crack had reached the extreme fibre of the flange, as illustrated in Figs. 4-6. One of the smallest cracks was discovered when examming a crack in the iillet weld. A tiny crack about 0.07~in. in diameter had originated from a very small pore in the left weld as shown in Fig. 4. The extent of the circular crack is seen from the smooth crack surface surrounding the pore, as compared to the rough appearance caused by static tearing when the cross section was opened for inspection. The small crack on the left was completely inside the Lange-to-web junction and could not be detected by inspection of the weld surface. A small crack discovered by the magnetic particle inspection method is shown in Fig. 5. The crack had initiated from the elongated pore and grown to the surface of the
Fatigue crack growth in welded beams
419
fillet weld. Almost no deviation from a circular shape is visible. This crack is about O-26in. diameter and could not be detected on the weld surface with the aid of a magnifying glass even under favorable circumstances and under sustained loading. A crack which has nearly penetrated the extreme fibre of the tension flange is shown in Fig. 6. Again, no significant disturbance of the circular shape is apparent at either side of the web or at the front approaching the surface of the flange. This crack, although quite sizeable, is not easily detected on a structural element under applied loading. The linear dimension of the crack visible on the surface of the flange-to-web junction is somewhat less than an inch. This phenomenon of a circular crack-shape,. seemingly not influenced by the free surfaces, is believed to be a consequence of the compatibility condition of the basically elastic cross section. Based on this photographic evidence, a circular disc-like crack was assumed to model the actual crack duriug growth inside the flange-web connection as was illustrated in Fig. 3. EVALUATION OF FATIGUR BRHAVIOR USING FiMTURR MECHANKS CONCEPTS The fatigue test data and fractographic observations of small fatigue cracks suggested that fracture mechanics of stable crack-growth might be useful in evaluating the observed behavior of the test beams. The stress intensity factor K introduced by Invin [ 111 describes in convenient form the influence of the stress, applied sufficiently away from the crack tip, and the crack size a. The relative size is usually expressed in terms of a correction function, f(u), where the linear dimensions of the plate, or the distance to a freeedge or surface, are introduced.
Paris [ 151 has suggested the following relationship between the rate of crack propagation and the change in the stress-intensity factor:
& ==CA&?.
(3)
This model expresses crack growth per cycle, d&N, in terms of the variation of the stress intensity factor, AK, and two material constants, C and n. For the case of a crack with constant correction factor, f(a), and subjected to constant amplitude stress, integration of equation (3) yields
where a! = (r&2)- 1. For specimens with equal initial crack-size Q, final crack-size ufi and identical boundary conditions, a theoretical value for the prediction of the life interval Nu can be expressed in terms of a new constant C’ times the applied stress EillgeAUas Nu =
C’Au-”
(5)
420
M. A. HIRT and J. W. FISHER
where
The log-log transformation of equation (5) yields a straight line of the form log N, = log C’ - n log (Au). AI’PLKATiOl’J
TO TEE PLAINWBLDED
(7)
BEAM
In order to apply the outlined concepts of fracture mechanics a reasonable characterization of the flaws was needed. A large number of cracks that origmated from porosity in the flllet weld were examined closely to establish the initial flaw condition. Flaws were photographed and enlarged to at least four times, as illustrated in Fig. 1.The dimensions of the defects were measured on the photos under lo-power magnification. Cracks were examined in beams fabricated from three diRerent grades of steel. The estimated K-values corresponding to the measured flaw dimensions are summarized in Fig. 7 in a log-log presentation of AK/Au vs the controlling crack-size a. The estimate for the defect that corresponds to the largest observed crack from each beam is indicated by a solid symbol. The estimated K-values were obtained using a circumscrii ellipse for flaws similar to the shapes illustrated in Fig. l(a) and Fig. 6. For flaws comparable to those shown in Fig, l(b) and Fig. -5, the K+stimate at the narrow transition from the circular void to the elongated pore was generally used. It was previously concluded from fractographic inspection of very small fatigue cracks (Figs. 4 and 5) that a penny-shaped crack-model described the shape of these cracks during crack growth. However, the estimated K-values for the initial flaws shown in Fig. 7 represent a collection of elliptical shapes with various &ratios. Hence, each individual defect was transformed into a penny-shaped crack with an
0.4-
0. I
0.01
A
A36
o
A441
I
I
1
I
0.02
0.03
0.04
0.05
CRACK
SIZE.
1
1
o. I”
Fig. 7. Distribution of estimated k-values and equivalent crack-radii.
I
1
0. IO
421
Fatigue crack growth in welded beams
equivalent crack-radius a, corresponding to the estimated AK/Ao-value. The sample of the resulting equivalent crack-radii is indicated at the bottom of Fig. 7. An average equivalent crack-radius a, = 0.04 in. was selected to represent the sample of measured pores. The initial crack-radius af was assumed equal to this average value. Derivation ofcrack-growth constants. The coefficients of the crack-growth equation can now be established from the equivalence of the coefficients in equations (1) and (7). Equation 7 is the theoretical prediction of the fatigue life using fracture mechanics concepts, and equation 1 expresses the statistical mean line of the relevant fatigue test data shown by solid dots in Fig. 2. The following assumptions were made to assist with the evaluation of the crack-growth constants, C and n: (1) The crack was assumed to be described by a disc-like penny-shaped crack with a constant correction factorf(u) over the interval of integration. The correction . factorsf(u) for a penny-shaped crack is 2/n= (2) The estimates of the initial and final crack-radii uj and uj were available from visual observations. The average initial crack-radius a, was assumed to equal the estimate of the equivalent crack-radius II, = O-04in. from Fig. 7 representing the sample of measured pores. The final crack-radius a, was assumed to be reached when the crack had penetrated the extreme fibre of the flange. The life remaining after this occurred was at most 10 per cent of the total fatigue life of the beam as was illustrated by the measurements given in Fig. 3. The final crack-radius was assumed to be equal to the nominal flange thickness. Experimental observations indicated this to be reasonable. (3) It was assumed that all three grades of steel could be represented by the same crack-growth equation. This assumption was based on results of tests on the welded beams. It was shown in Ref. 4 that grade of steel did not significantly influence the fatigue life of the beams. Equating the values of the coefficients for the mean regression curve [equation (111 n and C’ [equation (7)1, and substituting the crack-radii a, and a,, and the correction factorf(u) into equation (6) yields growth constants n a 3 and C = 2.05 X IO-*9 Substitution of these constants into equation 3 numerically describes the crackgrowth rate in terms of AK: to the coe&ients
$$ = 2.05 x lo-l”AK3.
W-t
COMPARISON OF DKRIVKD CRACK-GROWITI KQUATION WITR
MWSURRD CRACK-GROWTH RATES Equation 8 as derived from the mean regression curve fatigue test data is plotted in Fig. 8 and compared with the various stages of crack growth in the test beam illustrated in Fig. 3. The straight line estimate for crack growth as a penny-shaped crack from crack initiation until penetration of the extreme fibre is indicated. Also shown are the data points for the measured crack-growth rates on the inside and outside surfaces of the flange for growth as a three-ended crack.
of
IThe dimensions used for crack-growth rates, da/dN, were h/cycle, AK, was ksi 6.
and the stress-intensity factor range,
422
M. A. HIRT and J. W. FISHER
It is apparent from Fig. 8 and the schematic shown in Fig. 3 that most of the life is spent growing the crack from its initial equivalent flaw radius, a, = 0.04 in, to its penetration of the extreme fibre of the flange, a, = O-375in. The corresponding range of AK for the test beam was between 8 ksi 6 and 25 ksi V’& under a constant amplitude stress range of 36 ksi while 435,000 cycles elapsed. The transition from a penny-shaped crack to a three-ended crack only required about 16,000 cycles. An additional 10,000 cycles were required to grow the crack large enough to cause netsection yielding. Final failure terminated testing at 467,000 cycles. It is also apparent from Fig. 8 that most of the life was spent while growth occurred in a region of small AK. This is particularly true for lower applied stress ranges as illustrated in Fig. 9. The AK-regions applicable to growth as a penny-shaped crack are indicated for the test beams subjected to the stress ranges used in this study. Crack initiation took place at AK-values below 10 ksi 6. in all test beams. Most of the life was spent at growth rates smaller than lO+ in/cycle. The derived crack-growth relationship given by equation 8 is also compared with data from crack-growth specimens in Fig. 9. It was necessary to extrapolate the curve into the higher growth rate region to compare with available growth-rate data. The extrapolated curve falls within the scatterband reported by Crooker and Langel31. This scatterband contains data from tests on carbon and low-alloy steels with yield strengths between 34 ksi and 127 ksi. This is comparable to the yield strengths of the steel beams. A conservative upper bound for growth rates on ferrite-pearlite steels was proposed by Barsom [ 1I as
$$ = 3.6 x 10-‘“pK3.
(9)
This relationship is parallel to equation (8).
k
l
Crack on
Inside
A
Crackon
Extreme
Surface Fibfe
10-S
I
10-7
I
10-s CRACK-GROWTH
I
10-s RATE,
da/dN,
L
IO-' in
/cycle
Fig. 8. Various stages of crack growth and corresponding growth rates for a crack in a plainwelded beam starting from a pore in the flllet weld.
Fatigue crack growth in welded beams
423
Maddox (hf. 13)
E lJY
I
I 10-a
I IO-’
10-a
CRACK-GROW’TII
RATE.
I 10-s
I UP
doldN , in./CyEyclr
Fig. 9. Ranges for crack growth as a penny-shaped crack. Scatterbands and an upper bound estimate from other crack growth studies.
Since much of the growth in the plain-welded beams took place in the weld metal and heat affected zone, a comparison of the theo&& curve with Maddox’s data [ 133 is relevant. An approximate envelope for Maddox’s data on four diRerent weld metals is shown. Three had about equal yield strength (67 ksi), the fourth had a higher yield point equal to 90 ksi. Also included in the scatterband are test data for a simulated heat affected zone in mild steel material with the same yield strength. The correlation with these crack-growth data is good. It is apparent from Fig. 9 that the Croaker-Lange scatterband does not cover the critical region of interest for plainwelded beams. Most studies of crack-growth rates have been limited to larger AK-values and higher crack-growth rates because of the difficulties encountered in the slow growth region. However, the Woretical curve [Eq. @)I extrapolatedinto the higher AK-regions shows the same general trend reported by others on basic crack-growth specimens (Fig. 9). The &or&al curve is just above the crack-growth data and underestimates their growth rate. This is surprising since the penny-shaped crack assumes the best condition for the crack in the welded beams and neglects the influence of free surfaces. This underestimate in gt~~wthrate may be due to crack initiation or an overestimate of the stress intensity. The exponent, n, of the predicted crack-growth equation was about equal to 3-00 [Eq. @)I. It represents the slope of the fatigue test data on plain-weldedbeams fabricated from three grades of steel. Crooker and Lange [3] observed from a review of the literature that the value of the slope n fell between 2 and 4 for a large range of steels. Gurney [6] reported the slope of the curve to be a linear function of yield stress of the material. A general trend observed in crack-growth studies is for the value of the exponent, n, to decrease with increasing yield strength of the material. This trend was not as pronounced in the welded-beam study. The exponent n did not appear to vary greatly from 3. Careful evaluation of the coefficients n and C is needed for a wider range of rates of growth. Most of the data generally used to fit the straight line approximation only
424
M. A. HIRT and J. W. FISHER
extend over a small range of AK. In other cases, data points at the extremes cause rotation of the line. Substantial over- or under-estimates of the fatigue life of a structural component might result if crack-growth relationships are used to extrapolate beyond the range of the test data. Very slow growth. Johnson and Paris [ 121have suggested that a threshold exists for crack growth. A recent investigation by Paris [ 161on ASTM 93 10 steel has provided more information on this phenomenon of very slow growth. The test specimens were precracked and growth data for small AK-values obtained. The value of AK was then reduced stepwise and observations made on the relative crack-growth rates until a “threshold value” was reached. When AK was increased in steps, the rate of growth was comparable to the rates obtained by previous overloading. An approximate mean line fit to Paris’ data is compared in Fig. 10 with equation (8). A drastically reduced growth rate for AK = 5.2 ksi 6 is observed. Similar deviations from the straight line approximation are found for large AK-values indicating an increased growth-rate and apply mainly to low cycle fatigue problems. Harrison[7] has reviewed the literature for run-out data on a variety of specimens. He concluded that “for all materials with the exception of pure aluminum, cracks will not propagate if AK/E < 1 X 10-4&” He also found for a number of materials that the limit for non-propagating cracks fell between AK/E = 1.5 X lo44 and 1.8 X 10-l& Harrison’s levels for four types of steel are also shown in Fig. 10, and are , between 3.3 ksi & for mild steel and 5.3 ksi & for austenitic steel. Iqjlkence of defect-size on fatigue [email protected] life interval between two crack-sixes a( and a, can be determined from equation (4). For a given Iinal crack-size, the computed number of cycles can be expressed as a function of the initial crack-size for a given stress range, as shown by the curved lines in Fig. 11. The final crack-radius for the penny-shaped crack-model was assumed equal to the I&urge-thickness of the welded test beams. This becomes less important for larger differences between initial crackradius ai and final radius a, Alternately, the observed fatigue life of a tested beam can be used with the approp
“I
'10-s
10-e CRACK-GROWTH
10-p
lo-7
RATE,
do/dN,
10-5
in./cyti
Fig. 10. Approximate mean line of Paris’ data for very slow growth, and Harrison’s limiting AK-values for non-propagating cracks.
Fatigue crack growth in welded beams Stress
425
Ronpe, ksi
Mwaund , Qukoknt Flow - Radius %
1 2
NUMBER Fig.
I 1.
OF CYCLES
Cosnparism of the scatter of measured equivalent Aaw-sizes with t&e scatter of flawsizes derived ftom fktigue data.
riate stress-range line to estimate the equivalent initial pore-radius that caused faih,ue. The fatigue data of beams failing from porosity from Fig. 2 are replotted in this manner in Fig. 11. The resulting scatter of the equivalent pore-radii derived from the fatigue test data is shown on the right ordinate. The equivalent pore-radii determined independently from measured f&s are shown on~k~o~~eofF~ ll.Thtscrttteroftfiepwosi.tyasdtrivedfromthe~ test data (right ordinate) is seen to be similar to the scatter from the measured equivalent pore-sixes (left ordinate). Hence, if no variation is assumed in the crack-growth characteristics, this comparison suggests that the scatter in the fatigue test data is caused by the variation in the initial crack&e. This is essentially the same conclusion reached by Harrison [81for the analysis of test results from butt welds with lack of penetration defects. He concluded that the width of the scatterband probably resuhed from variations in initial radii at the defect tips. ~e~c~~ of furigue l&c using dwerertt cr~~-~e~. Additional crack-models were used to assess their influence on the prediction of the fatigue life. Figure 12 summa&es the results for the various models used. Six A441 steel beams were sekcted for the comparison. The actually observed fatigue lives of these six beams are shown by solid dots together with the mean and the 95% confidence interval for all test beams failing from porosity. The defects that caused crack growth and f’ailure of the six beams were examined. They were assumed to be characterixed by a circumscrii eilipse with half-axes a and b. The following models were used to estimate the fatigue life of each individual beam using the known dimensions of the circumscribed ellipse, and empioying crack-growth equation (8): Model (a) me open circles in Fig. 12 correspond to the life estimates for the penny-shaped crack with equivalent crack-radius. The equivalent crack-radius was computed from the equivalence of the K-value with the circumscribed elliptical crack.
M. A. HIRT and J. W. FISHER
426
/ /
Mean Life and ConfidenceInterval from 011 Welded Beams Failinq from Porosity (Fig. 2 1
(a)
o Equivalent Cmch - Size
(b)
I- Circumscribed and -( Inscribed Circle
Made1 (c) A Elliptical Crock-Model ‘L,-’
51 IO’
l I1
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Observed Fotiaue Life 1
5
I
.
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Fig. 12. Prediction of fatigue life using various crack-models.
Model (b) Penny-shaped cracks were assumed to inscribe and circumscribe the ellipse. The shortest lie estimates resulted from the circumscribed crack and the longest from the inscribed crack. These two estimates constitute “upper and lower bounds” and are indicated by the horizontal T-symbols in Fig. 12. Model (c) Additional life estimates were made based on the elliptical crack-model. Growth was assumed in the direction of the minor axis 2(a) with constant major axis 2(b) until the size of the circumscribed crack was reached. The computation of the increment of life from an elliptical to a circular crack was done using an available computer program[5]. Numerical integration was employed because of the variable correction factor f(a). This increment of life ivas added to the estimate for the circumscribed penny-shaped crack. The total life estimate is indicated by triangles in Fig. 12. The comparison between the estimates for each individual beam provided by the different models and the observed fatigue life permitted the following observations to be made: (1) The life estimates employing the equivalent penny shaped crack-radius [model (a)] resulted in slightly shorter lives than the estimates that assumed the elliptical crack first to grow to a circular shape [model (c)l. (2) Both of these estimates were bounded by the estimates provided by the inscribed and circumscribed initial cracks [model (b)]. (3) The observed fatigue data were also contained within the bounds from the estimates of the circumscribed and inscribed circles in all but one case. (4) Long flaws with small a/b-ratios [as illustrated in Fig. l(b)] provided the greatest deviation between the computed “lower and upper bounds.” (5) All three defects of the type shown in Fig. l(a) had their life underestimated when the equivalent initial crack-radius was used [model (a)]. The estimate improved for the elliptical model (c). (6) Most of the estimates from models (a) and (c) fell within the two limits of disper-
427
Fatigue crack growth in welded beams
sion representing the 95 per cent confidence interval of all the beam test data. The selection of a penny-shaped crack model did not introduce more scatter than was experimentally observed from fatigue test data Prediction of fatigue life using measured crack-growth rates. Maddox[l4] and Harrison [8] have demonstrated that good predictions ofthe fatigue life can be obtained from crack-growth data. Barsom’s[l] conservative estimate of crack-growth rate, equation (91, for ferrite-peariite steels was used to predict the fatigue life of the beams failing from porosity. The result is shown in Fig 13 together with the applicable welded beam test data and the respective mean line and the 95 per cent confidence interval. Also shown in Fig. 13 is the prediction based on Paris’ data [ 161,indicating a threshold level for run-out at about 23 ksi stress range. This prediction was obtained using the straight line segments shown in Fig. 10 which approximate the data from the study on very slow growth. The threshold values by Harrison[7] shown in Fig. 10 were also used to estimate run-out. Run-out tests are predicted to occur at levels as high as 20 ksi stress range for low-alloy steel, as shown by the horizontal lines in Fig. 13. The predictions based on the crack-growth data underestimate the mean fatigue life of the test beams at the higher stress range levels. This was expected from the comparison of the crack-growth data Since any assumption other than a penny-shaped crack-model for the welded beams would further reduce the life prediction based on crack-growth data, a crack-initiation period may be responsible for the slight underestimate. Other factors that could be responsible for the underestimate in life are an overestimate of the initial crack-size, an overestimate of the stress intensity, or a slower growth rate under plane-strain conditions [2]. ThescatterinthetestdatashowninFig. 13waspr&ouslyrelatedtothevariation inshape,sizeandseverityoftheporosity.othtrfactorscontributetothevariationin test data. Beams with seemingly equal defects still experience variation in the fatigue strength because of variation in the crack tip radius, crack-growth rates, and other uncontrolled variables. 100
r 95 K
Estimota
Confidence lntrrvol
from Growth Dota
From “Threshold” (Fig. IO) Low-Alloy
StnlJ
Mild Stool 5105
NUMBER Fig. 13. Comparison
I ~.~~I
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IO’
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OF CYCLES
of a pndictcd lower bound and a mean line with mean life and scatterband from test data.
428
M. A. HIRT and J. W. FISHER
SUMMARY AND CONCLUSIONS The main findings of this study are summarized hereafter. They are based on a detailed examination of the test data, fracture surfaces, initial flaw conditions provided by the experimental work, and on the theoretical studies of stable crack growth. (1) Fractographic e~on of the initial flaw conditions revealed porosity to be the most common defect in plain-welded beams. The distribution of the size and location of the pores in the longltudlnal fillet weld was random. The welded beam fatigue data fell within a narrow scatterband when plotted as the logarithmic transformation of stress range and cycles to failure. (2) Fracture mechanics concepts provided a rational way to analyze and characterize the behavior of welded (and rolled) beams. These concepts were applied to describe numerically the “efktive” initial Saw conditions in welded beams, and to derive a crack-growth rate vs range of stress-intensity relationship from welded-beam fatigue data. (3) A penny-shaped crack was found to model crack growth from porosity in welded beams. An equivalent crack with a GO4in radius described the average pores observed in the welded beams. (4) The derived crack-growth equation exhibited the same trend as measured data from crack-growth specimens. It provided a lower estimate of the growth rate. Among other factors, this dl@erence was attributed to crack axon, a possible overestimate of the stress intensity, and slower growth in the piane-strain condition for the welded beams. (5) Available crack-growth data were shown to only cover a small region of growth rates. Extrapolation into regions outside the data could be misleading, particularly when used for fatigue life estimates. (6) Very little information is available for growth rates below 1W8in./cycle. This was found to be the region most critical for the fatigue behavior of welded and rolled beams. More than 75 per cent of the life was spent in this region growing a crack from its initial size to a visible crack. (7) It was shown that the initial flaw-size was the controlling factor for the fatigue life of welded beams. An increase in flange-thickness and larger weld-sizes should hence not permit an increase in allowable defect-size. (8) The variation of the fatigue data within the scatterband was found to be related in part to the variation in fatigue crack-growth rates or other uncontrolled variables. ~c&~~fe~ge~~f~-~ paper is based on the cxperimentai and theoretical ~v~~ns made during the course of a research progmm on the &cct of weidmcnts on the fatigue strength of steel beams. Partial support was also provided by the tXIc.c of Naval Research, Dcpartmmn of Defense, under Contract NO001468-A-514; NRO64-509. The expmimental research was performed under National Cooperative Highway Research Prngram Project 12-i’. The opinions and 6ndings expnssed or implied in this paper are those of the authors. They arc not necessarily those of the ffighway Reset&r Board, the National Academy of Sciences, the Federal Highway Administration, the Amatican Association of State Highway Gfiiciais, nor of the individual states participating in the NCHRP Program. The authors wish to express their appreciation for the helpful suggestions made by Profs. G. R. Irwin, A. W. Pcnsc. and R. W. Hertzberg on the fracture mechanics and metaUurgicat aspects of this work. Special thanks are due Karl H. Frank and Ben T. Yen for the cooperation and chaBenging discussion during the progmss of this study,
REFJXRENCES [ll J. M. Barsom, Fatigue-crack propagation in steels of various yield strengths, U.S. Steel Corp., Applied Research Laboratory, Monrocvihe, Pa.. ( 197 I ). 121 W. G. Jr. Clark, and H. E. Jr. Trout, InfIuence of temperature and section size on fatigue crack growth behavior in Ni-Mo-V Alloy Steel. Engng Frucrure Me&. 2. 107-123 (1970).
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[3] T. W. Crookcr and E. A. Lange, How yield strength and fracture toughness considerations can influence fatigue design procedures for structural steels, Welding Research Supplement, 49, (10) 488-496 October, (1970). [41 J. W. Fisher, K. H. Frank, hf. A. Hirt and B. M. McNamee, Effect of weldments on the fatigue strength of steel beams, NCHRP Report No. 102, Highway Research Board, National Academy of SciencesNational Research Council, Washington, D.C., (1970). [51 K. H. Frank and J. W. Fisher, Analysis of error in determhnng fatigue crack growth rates, Report No. 358-10, Lehigh University, Fritz Engineering Laboratory, March, (1971). 161 T. R. Gurney, An investigation of the rate of propagation of fatigue cracks in a range of steels, Members’ Report No. E18/12/68, The Welding Institute, December, (1968). PI J. D. Harrison, An analysis of data on non-propagating fatigue cracks on a fracture mechanics basis, Metal Construction and Br. Weld. .I. 2.93-98 (1970). 181 J. D. Harrison, The analysis of fatigue test results for butt welds with lack of penetration defects using a fracture mechanics approach, Weld. World, 8,168- 18 1 ( 1970). 191 M. A. Hirt, B. T. Yen, and J. W. Fisher, Fatigue strength of rolled and welded steel beams, J. Srruc. Diu. ASCE,97,1897-1911(1971). [lo] M. A. Hit?, Fatigue behavior of roiled and welded beams, Ph.D. Dissertation, Lehigh University, Department of Civil Engineering, October, ( 197 1). [ 11) G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Transactions, ASME, Series E, Vol. 24, (3) 361-364 (1957). [12] H. H. Johnson and P. C. Paris, Sub-critical flaw growth, Engng Fracture Mech. 1, (1) 3-45 (1968). 113) S. J. Maddox, Fatigue crack propagation in weld metal and heat affected zone material, Members’ Report No. E/29/69, The Welding Institute, December, (1969). I.141 S. J. Maddox, The propagation of part-through-thickness fatigue cracks analysed by means of fracture mechanics, Members’ Report No. E/30/69, The Welding Institute, December, (1969). [15] P. C. Paris, The fracture mechanics approach to fatigue, Fatigue-An interdisciplinary Approach, Proc. Tenfh Sagamore Army Materials Research Cot&, Syracuse University Press, Syracuse, N.Y., p. 107 (1964). 1161 P. C. Paris, Testing for very slow growth of fatigue cracks, Closed Loop, MTS Systems Corp., 2, 5, I l-14 (1970). (Received 4 April 1972)